{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:08:25Z","timestamp":1783904905672,"version":"3.55.0"},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2018,4,1]],"date-time":"2018-04-01T00:00:00Z","timestamp":1522540800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    Tim Makarios (with Isabelle\/HOL\n                    <jats:sup>1<\/jats:sup>\n                    ) and John Harrison (with HOL-Light\n                    <jats:sup>2<\/jats:sup>\n                    ) shown that \u201cthe Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski\u2019s axioms except his Euclidean axiom\u201d [3], [4], [14], [5].\n                  <\/jats:p>\n                  <jats:p>With the Mizar system [2], [7] we use some ideas are taken from Tim Makarios\u2019 MSc thesis [13] for the formalization of some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as further development of Tarski\u2019s geometry in the formal setting [6]. Note that the model presented here, may also be called \u201cBeltrami-Klein Model\u201d, \u201cKlein disk model\u201d, and the \u201cCayley-Klein model\u201d [1].<\/jats:p>","DOI":"10.2478\/forma-2018-0003","type":"journal-article","created":{"date-parts":[[2018,8,2]],"date-time":"2018-08-02T20:32:23Z","timestamp":1533241943000},"page":"21-32","source":"Crossref","is-referenced-by-count":7,"title":["Klein-Beltrami Model. Part I"],"prefix":"10.2478","volume":"26","author":[{"given":"Roland","family":"Coghetto","sequence":"first","affiliation":[{"name":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2018,7,28]]},"reference":[{"key":"2026071213090489846_j_forma-2018-0003_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","unstructured":"[1] Norbert A\u2019Campo and Athanase Papadopoulos. On Klein\u2019s so-called non-Euclidean geometry. arXiv preprint arXiv:1406.7309, 2014."},{"key":"2026071213090489846_j_forma-2018-0003_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071213090489846_j_forma-2018-0003_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","unstructured":"[3] Eugenio Beltrami. Saggio di interpetrazione della geometria non-euclidea. Giornale di Matematiche, 6:284\u2013322, 1868."},{"key":"2026071213090489846_j_forma-2018-0003_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Eugenio Beltrami. Essai d\u2019interpr\u00e9tation de la g\u00e9om\u00e9trie non-euclid\u00e9enne. In Annales scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure. Trad. par J. Ho\u00fcel, volume 6, pages 251\u2013288. Elsevier, 1869.10.24033\/asens.60","DOI":"10.24033\/asens.60"},{"key":"2026071213090489846_j_forma-2018-0003_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[5] Karol Borsuk and Wanda Szmielew. Podstawy geometrii. Pa\u0144stwowe Wydawnictwo Naukowe, Warszawa, 1955 (in Polish)."},{"key":"2026071213090489846_j_forma-2018-0003_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] Adam Grabowski. Tarski\u2019s geometry modelled in Mizar computerized proof assistant. In Proceedings of the 2016 Federated Conference on Computer Science and Information Systems, FedCSIS 2016, Gda\u0144sk, Poland, September 11\u201314, 2016, pages 373\u2013381, 2016. doi:10.15439\/2016F290.10.15439\/2016F290","DOI":"10.15439\/2016F290"},{"key":"2026071213090489846_j_forma-2018-0003_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa","doi-asserted-by":"crossref","unstructured":"[7] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026071213090489846_j_forma-2018-0003_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa","unstructured":"[8] Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381\u2013383, 2003."},{"key":"2026071213090489846_j_forma-2018-0003_ref_009_w2aab3b7b2b1b6b1ab1ab9Aa","unstructured":"[9] Artur Korni\u0142owicz and Yasunari Shidama. Intersections of intervals and balls in \ud835\udca0Tn${\\cal E}_T^n$. Formalized Mathematics, 12(3):301\u2013306, 2004."},{"key":"2026071213090489846_j_forma-2018-0003_ref_010_w2aab3b7b2b1b6b1ab1ac10Aa","unstructured":"[10] Artur Korni\u0142owicz and Yasunari Shidama. Some properties of circles on the plane. Formalized Mathematics, 13(1):117\u2013124, 2005."},{"key":"2026071213090489846_j_forma-2018-0003_ref_011_w2aab3b7b2b1b6b1ab1ac11Aa","unstructured":"[11] Akihiro Kubo. Lines in n -dimensional Euclidean spaces. Formalized Mathematics, 11(4): 371\u2013376, 2003."},{"key":"2026071213090489846_j_forma-2018-0003_ref_012_w2aab3b7b2b1b6b1ab1ac12Aa","doi-asserted-by":"crossref","unstructured":"[12] Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. Formalized Mathematics, 18(1):1\u201310, 2010. doi:10.2478\/v10037-010-0001-2.10.2478\/v10037-010-0001-2","DOI":"10.2478\/v10037-010-0001-2"},{"key":"2026071213090489846_j_forma-2018-0003_ref_013_w2aab3b7b2b1b6b1ab1ac13Aa","unstructured":"[13] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski\u2019s Euclidean Axiom. Victoria University of Wellington, New Zealand, 2012. Master\u2019s thesis."},{"key":"2026071213090489846_j_forma-2018-0003_ref_014_w2aab3b7b2b1b6b1ab1ac14Aa","unstructured":"[14] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535\u2013545, 1991."},{"key":"2026071213090489846_j_forma-2018-0003_ref_015_w2aab3b7b2b1b6b1ab1ac15Aa","unstructured":"[15] Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541\u2013547, 2005."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/26\/1\/article-p21.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2018-0003","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:26:14Z","timestamp":1783902374000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2018-0003"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,1]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,7,28]]},"published-print":{"date-parts":[[2018,4,1]]}},"alternative-id":["10.2478\/forma-2018-0003"],"URL":"https:\/\/doi.org\/10.2478\/forma-2018-0003","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2018,4,1]]}}}